Functional inequalities for the heat flow on time-dependent metric measure spaces

Abstract

We prove that synthetic lower Ricci bounds for metric measure spaces -both in the sense of Bakry-Émery and in the sense of Lott-Sturm-Villani -can be characterized by various functional inequalities including local Poincaré inequalities, local logarithmic Sobolev inequalities, dimension independent Harnack inequality, and logarithmic Harnack inequality.More generally, these equivalences will be proven in the setting of time-dependent metric measure spaces and will provide a characterization of super-Ricci flows of metric measure spaces.

Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometry

“…Namely, (M, g(t)) t∈I is called super Ricci flow if ∂ t g ≥ −2 Ric, which has been introduced by McCann-Topping [33] from the viewpoint of optimal transport theory. Recently, the super Ricci flow has begun to be investigated from various perspectives, especially metric measure geometry (see e.g., [3], [4], [16], [19], [20], [21], [25], [26], [27], [28], [29], [30], [39]). A Ricci flow (M, g(t)) t∈I is said to be ancient when I = (−∞, 0], which is one of the crucial concepts in singular analysis of Ricci flow.…”